American Option Pricing with Binomial Tree

The buyer of an American Option has the right to exercise the option at any time before and including the maturity date of the option while the buyer of a European Option has the right to exercise the option at the maturity date only.

We assume that there is no arbitrage and we place ourself in discrete time in the Cox-Ross-Rubinstein binomial model framework.

We know that in this framework the price of an European option is equal to the discounted expectation of its final payoff under the risk-neutral probability Q.

We don’t have such simple expression for American options.

But we can calculate the price of an American option with a backward recursion.

The price of the option at maturity is equal to its final payoff. At the previous dates, the option’s holder will decide to exercise the option if it is interesting for him, so if the value of exercising the option is higher than the value of keeping it.

So the price of the American option at a given date t is equal to the maximum between the value of exercising it at t, and the discounted expectation of the value of the option at the next date.

The underlying asset price can go up by a factor u with a probability q and down by a factor d with a probability 1 – q. We know the expression of the risk-neutral probability q, and the upward and downward factors u and d are determined from the volatility sigma of the underlying asset price.

So we can determine the price of the American option starting from the boundary condition with the final payoff of the option and applying a recursive backward algorithm.

An American option, which gives more rights than a European option, has a higher price. It can be easily shown by using the backward recursion formula for the American option price, and the fact that the European option price is equal to the discounted expectation of the future price.

For call options, the price of American and European options are the same if we assume no dividend payments and a positive risk free interest rate. There is indeed no benefit for the American call option holder to exercise the option before it expires. The price of the option is always higher than the exercise price.

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List of Quant Next Courses

We summarize below quantitative finance training courses proposed by Quant Next. Courses are 100% digital, they are composed of many videos, quizzes, applications and tutorials in Python.

Complete training program:

Options, Pricing, and Risk Management Part I: introduction to derivatives, arbitrage free pricing, Black-Scholes model, option Greeks and risk management.

Options, Pricing, and Risk Management Part II: numerical methods for option pricing (Monte Carlo simulations, finite difference methods), replication and risk management of exotic options.

Options, Pricing, and Risk Management Part III:  modelling of the volatility surface,  parametric models with a focus on the SVI model, and stochastic volatility models with a focus on the Heston and the SABR models.

A la carte:

Monte Carlo Simulations for Option Pricing: introduction to Monte Carlo simulations, applications to price options, methods to accelerate computation speed (quasi-Monte Carlo, variance reduction, code optimisation).

Finite Difference Methods for Option Pricing: numerical solving of the Black-Scholes equation, focus on the three main methods: explicit, implicit and Crank-Nicolson.

Replication and Risk Management of Exotic Options: dynamic and static replication methods of exotic options with several concrete examples.

Volatility Surface Parameterization: the SVI Model: introduction on the modelling of the volatility surface implied by option prices, focus on the parametric methods, and particularly on the Stochastic Volatility Inspired (SVI) model and some of its extensions.

The SABR Model: deep dive on on the SABR (Stochastic Alpha Beta Rho) model, one popular stochastic volatility model developed to model the dynamic of the forward price and to price options.

The Heston Model for Option Pricing: deep dive on the Heston model, one of the most popular stochastic volatility model for the pricing of options.